Chapter 6 - Your First Four Derivatives

In this chapter I'm going to present the four most important rules of derivatives that you must know and memorize. These rules come from logical results and calculations of function limits. Mathematicians have used function limits to come up with these rules so you can find derivatives for a massive number of functions quickly and easily. Knowing and memorizing these formulas is easy. Let's get started.

Rule 1: The Derivative of a Constant
Let's assume that $C$ is a constant number like 2, 17, or 3.141592. The derivative of a constant is zero. In other words, \[ \frac{d}{dx}C=0. \] Remember that a derivative is a slope (rise over run). So how much does a constant "rise" when you run? It doesn't, and that's why the derivative is zero.

Rule 2: The Derivative of a Constant Times a Function
Let's assume $C$ is a constant number. If $f(x)$ has the derivative $df/dx$, then the derivative of $C\cdot f(x)$ is \[ \frac{d}{dx}\Big[C\cdot f(x)\Big]=C\cdot \frac{df}{dx}. \] And here it is in prime notation: \[(C\cdot f)\,'(x)=C\cdot f\,'(x).\]

Rule 3: The Derivative of the Sum of Two Functions

Let's assume that $h(x)$ is the sum of two functions $f(x)$ and $g(x)$. That is, $h(x)=f(x)+g(x)$. If $f(x)$ has the derivative $df/dx$ and $g(x)$ has the derivative $dg/dx$, then the derivative of $h(x)$ is \[\frac{d}{dx}h(x)=\frac{d}{dx}\Big[f(x)+g(x)\Big]=\frac{df}{dx}+\frac{dg}{dx}\] In other words, just add the derivatives of the individual functions. Somtimes it's helpful to see it in prime notatino. Here it is: \[h\,'(x)=(f+g)\,'(x)=f\,'(x)+g\,'(x)\]

You can now combine these in obvious and cool ways.

For example, what if you wanted to know the derivative of $f(x)-g(x)$? Think of $-g(x)$ as $C\cdot g(x)$ where $C=-1$. This means you need to find the derivative of $f(x)+C\cdot g(x)$. From Rule 3 you know that

\[\frac{d}{dx}\Big[f(x)+C\cdot g(x)\Big]=\frac{df}{dx}+\frac{d}{dx}\Big[C\cdot g(x)\Big].\]

And from Rule 2 you know that

\[\frac{d}{dx}\Big[C\cdot g(x)\Big]=C\cdot \frac{dg}{dx}.\]

Plugging this back in you have

\[\frac{d}{dx}\Big[f(x)+C\cdot g(x)\Big]=\frac{df}{dx}+C\cdot\frac{dg}{dx}.\]

Finally, plug in $-1$ for $C$ to get

\[\frac{d}{dx}\Big[f(x)-g(x)\Big]=\frac{df}{dx}-\frac{dg}{dx}.\]

With Rules 2 and 3 you discovered the derivative of the subtraction of two functions!

Okay, now for one of the most important rules Calculus - The Power Rule.

Rule 4: The Power Rule
\[\frac{d}{dx}x^n=nx^{n-1}\] where $n$ is an integer. (Remember an integer is a number like 1,3 or -100, but not 1.5 or 2.3.)

For example, the Power Rule means the derivative of $x$ is $1$, the derivative of $x^2$ is $2x$, and the derivative of $x^3$ is $3x^2$. You have to know and memorize the Power Rule!

What if you want to find the derivative of $4x^3$? Think of this as $C\cdot f(x)$ where $C=4$ and $f(x)=x^3$. Rule 2 says that

\[\frac{d}{dx}\Big[C\cdot f(x)\Big]=C\cdot \frac{df}{dx}.\]

This means that

\[\frac{d}{dx}\Big[4\cdot x^3\Big]=4\cdot \frac{d}{dx}x^3,\]

but from the Power Rule

\[\frac{d}{dx}x^3=3x^2.\]

This means that

\[\frac{d}{dx}\Big[4\cdot x^3\Big]=4\cdot 3\cdot x^2=12x^2.\]

The exercise above is the basis for rule 5:

Rule 5: The Derivative of $C\,x^n$
\[\frac{d}{dx}C\,x^n=Cnx^{n-1}\] where $n$ is an integer and $C$ is a constant. Remember, this comes from rules 2 and 4 so you only need to keep the first four rules in your head!

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Next, we'll take a brief aside so you can learn more about the tangent line. We'll get its exact formula and see how it can be used to approximate the value of a function.

Keep reading . . .